Hostname: page-component-5c6d5d7d68-lvtdw Total loading time: 0 Render date: 2024-08-18T20:17:11.975Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional absolute and convective instabilities at the onset of convection in a porous medium with inclined temperature gradient and vertical throughflow

Published online by Cambridge University Press:  25 November 2009

LEONID BREVDO
LEONID BREVDO*
Affiliation:
Institute of Fluid and Solid Mechanics, University of Strasbourg/CNRS, 2 rue Boussingault, F 67000 Strasbourg, France
*
†Email address for correspondence: Leonid.Brevdo@imfs.u-strasbg.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

By using the mathematical formalism of absolute and convective instabilities, we study in this work the nature of unstable three-dimensional localized disturbances at the onset of convection in a flow in a saturated homogeneous porous medium with inclined temperature gradient and vertical throughflow. It is shown that for marginally supercritical values of the vertical Rayleigh number Rv the destabilization has the character of absolute instability in all the cases in which the horizontal Rayleigh number Rh is zero or the Péclet number Qv is zero. In all the cases in which Rh and Qv are both different from zero, at the onset of convection the instability is convective. In the latter cases, the growing emerging disturbance has locally the structure of a non-oscillatory longitudinal roll, and its group velocity points in the direction opposite the direction of the applied horizontal temperature gradient, i.e. parallel to the axis of the roll. The speed of propagation of the unstable wavepacket increases with Qv and generally increases with Rh.

JFM classification

Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 641 , 25 December 2009 , pp. 475 - 487
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bahloul, A., Boutana, N. & Vasseur, P. 2003 Double-diffusive and Soret-induced convection in a shallow horizontal porous layer. J. Fluid Mech 491, 325–352.CrossRefGoogle Scholar
Bear, J. 1988 Dynamics of Fluids in Porous Media. Dover.Google Scholar
Brevdo, L. 1991 Three-dimensional absolute and convective instabilities, and spatially amplifying waves in parallel shear flows. Z. Angew. Math. Phys. 42, 911–942.CrossRefGoogle Scholar
Brevdo, L. & Ruderman, M. S. 2009 a On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part I. Normal modes. Transp. Porous Media. 80, 137–151.CrossRefGoogle Scholar
Brevdo, L. & Ruderman, M. S. 2009 b On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part II. Absolute and convective instabilities, and spatially amplifying waves. Transp. Porous Med. 80, 153–172.CrossRefGoogle Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Delache, A. & Ouarzazi, M. N. 2008 Weakly nonlinear interaction of mixed convection patterns in porous media heated from below. Intl J. Therm. Sci. 47, 709–722.CrossRefGoogle Scholar
Delache, A., Ouarzazi, M. N. & Combarnous, M. 2007 Spatio-temporal stability analysis of mixed convection flows in porous media heated from below: comparison with experiments. Intl J. Heat Mass Transfer 50, 1485–1499.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Gaster, M. 1968 The development of three-dimensional wave packets in a boundary layer. J. Fluid Mech. 64, 654–665.Google Scholar
Gaster, M. 1975 A theoretical model of a wave packet in the boundary layer on a flat plate. Proc. R. Soc. Lond. A 347, 271–289.Google Scholar
Gaster, M. & Grant, I. 1975 A theoretical model of a wave packet in the boundary layer on a flat plate. Proc. R. Soc. Lond. A 347, 253–269.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Narayana, P. A. L., Murthy, P. V. S. N & Gorla, R. S. R. 2008 Soret-driven thermo-solutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. J. Fluid Mech 612, 1–19.CrossRefGoogle Scholar
Nield, D. A. 1990 Convection in a porous medium with inclined temperature gradient and horizontal mass flow. In Heat Transfer 1990: Proceedings of the Ninth International Heat Transfer Conference, Jerusalem (ed. Hetsroni, G.), vol. 5, pp. 153–158. Hemisphere.Google Scholar
Nield, D. A. 1991 Convection in a porous medium with inclined temperature gradient. Intl J. Heat Mass Transfer 34, 87–92.CrossRefGoogle Scholar
Nield, D. A. 1994 Convection in a porous medium with inclined temperature gradient: additional results. Intl J. Heat Mass Transfer 37, 3021–3025.CrossRefGoogle Scholar
Nield, D. A. 1998 Convection in a porous medium with inclined temperature gradient and vertical throughflow. Intl J. Heat Mass Transfer 41, 241–243.CrossRefGoogle Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media. Springer.Google Scholar
Nield, D. A., Manole, D. M. & Lage, J. L. 1993 Convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. J. Fluid Mech 257, 559–584.CrossRefGoogle Scholar
Ouarzazi, M. N., Mejni, F., Delache, A. & Labrosse, G. 2008 Nonlinear global modes in inhomogeneous mixed convection flows in porous media. J. Fluid Mech 595, 367–377.CrossRefGoogle Scholar
Persson, A. O. 2006 Hadley's principle: understanding and misunderstanding the trade winds. History Meteorol. 3, 17–42.Google Scholar
Qiao, Z. & Kaloni, P. N. 1997 Convection in a porous medium induced by an inclined temperature gradient with mass flow. Trans ASME J. Heat Transfer 119, 366–370.CrossRefGoogle Scholar
Straughan, B. 2004 a Resonant penetrative convection. Proc. R. Soc. Lond. A 260, 2913–2927.CrossRefGoogle Scholar
Straughan, B. 2004 b The energy method, stability and nonlinear convection. Springer.CrossRefGoogle Scholar
Straughan, B. & Walker, D. W. 1996 Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. J. Comput. Phys. 127, 128–141.CrossRefGoogle Scholar
Twiss, R. Q. 1951 On oscillations in electron streams. Proc. Phys. Soc. Lond. B 64, 654–665.CrossRefGoogle Scholar
Weber, J. E. 1974 Convection in a porous medium with horizontal and vertical temperature gradients. Intl J. Heat Mass Transfer 17, 241–248.CrossRefGoogle Scholar